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A Multi-Objective Self-Adaptive Differential

Evolution Algorithm for Conceptual High-Rise

Building Design

Abstract— This paper presents a multi-objective self-adaptive

differential evolution algorithm to solve the form-finding problem

of high-rise building design in the conceptual phase. The aim of

the research is to reach suitable high-rise design alternatives for

hard and soft objectives, which are construction cost per square

meter, structural displacement, and visual perception of the spaces

from the inside out subject to several constraints that are related

with both high-rise construction regulations, and profitability of

the spaces. We formulate the problem as a multi-objective real-

parameter constrained optimization problem for three objectives

that are inherently conflicting. To tackle this problem, we

developed two different optimization algorithms, namely, a Non-

Dominated Sorting Genetic Algorithm II (NSGA-II) and a Self-

Adaptive Differential Evolution Algorithm (jDE) in order to

obtain Pareto fronts with diversified non-dominated solutions.

The extensive computational results show that the jDE algorithm

yields much more desirable Pareto front than the NSGA-II

algorithm.

Keywords— Evolutionary computation, computational design,

high-rise optimization, performance-based design, multi-objective

optimization.

I. I

NTRODUCTION

Architectural design is considered as one of the most

complex tasks. The main reason behind it is that architectural

design entails multiple objectives that are naturally conflicting

with each other. In addition, these objectives strongly influence

how to shape final solutions for architectural design. Therefore,

one of the most important roles of the architect is to satisfy both

hard and soft aspects of architectural design by means of

maximizing advantages and minimizing disadvantages starting

from the conceptual phase [1]. In the field of architecture, high-

rise buildings could be the best example of the most complex

structures to design and construct.

For the sake of built environment, the importance of those

buildings for future has been debated in the scientific

community for decades [2-5]. The world population is

constantly increasing, that is projected from 7 billion to reach to

9 billion [6]. Just 20 years ago, one third of the world population

was living in urban areas. However, now this figure has

increased to 50% and it is estimated to be more than 60% in 2030

[7]. One of the most important roles of high-rise buildings for

the environment is to support spaces in the limited physical land

area. In other words, high-rise buildings are able to deal with

dense populations in severely limited urban areas. It is thus clear

that urbanized cities will require more high-rise building

alternatives in the built environment. Regarding to land

consumption, high-rise buildings can provide much more open

spaces in the urban areas. This is also beneficial for the

preservation of natural areas around the cities, recreational

spaces, critical environmental areas and so forth [8]. In other

words, when cities expand vertically, places such as public

spaces, agricultural lands, and wilderness remain and so on can

stay untouched. For instance, accommodating the same number

of people in a high-rise building of forty stories, versus two

stories, requires about one-twentieth of the land. This suggests

an increasing demand to high-rise buildings that will play a

significant role for future urbanization.

This paper addresses a method to discover satisfactory

results for high-rise form finding design problem in the

conceptual design phase. Conceptual design is accepted as the

most critical design stage that affects all subsequent phases of

the design process. The aim of the research is to reach suitable

high-rise design alternatives for hard and soft objectives, which

are construction cost per square meter, structural displacement,

and visual perception of the spaces from the inside out, subject

to several constraints that are related with both high-rise

construction regulations, and profitability of the spaces. As our

main problem, we are considering the scenario of a new high-

rise design in Izmir, a city located in the west part of Turkey.

ଵ

,

ଶ

,

ଷ

Yasar University, TU Delft

Department of Architecture

Chair of Design Informatics

Izmir, Turkey, Delft, The

Netherlands

{berk.ekici, i.chatzikonstantinou,

i.s.sariyildiz}@yasar.edu.tr,

{i.chatzikonstantinou,

i.s.sariyildiz}@tudelft.nl

Ǥ

ସ

Yasar University

Department of Industrial

Engineering

Izmir, Turkey

fatih.tasgetiren@yasar.edu.tr

െ

ହ

State Key Laboratory,

Huazhong University of

Science and Technology,

Wuhan, PR China

panquanke@qq.com

2272

978-1-5090-0623-6/16/$31.00 c

2016 IEEE

Izmir has been experiencing and emerging in terms of

settlements for commercial developments over the past years,

focusing mainly on high-rise buildings. We formulate the

problem as a multi-objective real-parameter constrained

optimization problem for the objectives mentioned above, that

are inherently conflicting subject to several design constraints.

To cope with this problem we employed two different

optimization algorithms namely Non-Dominated Sorting

Genetic Algorithm II (NSGA-II) and Self-Adaptive Differential

Evolution Algorithm (jDE) that are capable to deal with multi-

objective real-parameter constrained optimization problems.

II.

PROBLEM DEFINITION

As mentioned before, three objectives are considered as

minimization of displacement value of the structure,

minimization of construction cost per square meter, and visual

perception of outside from indoor spaces.

Hence, discovering suitable solution alternatives by

considering these three objectives is a complex task in the

conceptual phase of high-rise design process. Furthermore, there

are also several constraints, which further increase the

complexity of the design problem, should be satisfied.

A. Objective Functions and Constraints

To discover a high-rise design that presents satisfactory

results for each objective, we consider this study as multi-

objective real-parameter constrained optimization problem by

focusing on the conceptual phase of the design process as

follows:

(

)

vpCvMin

csq

−,,

(1)

where

KFv ÷=

(2)

TOFACC

tcsq

÷=

cos

(3)

¸

¹

·

¨

©

§−

=FC

SAFA

vp

(4)

Subject to:

2

000.90 mTOFA ≤ (5)

500÷≤ Hv (6)

mFH 00.3≥ (7)

mLSm 0.1650.7 ≤≤

(8)

Regarding the first objective function, v is the displacement

of the high-rise structure,

F

is the forces implemented on each

high-rise alternative generated by the optimization algorithms

K

is the stiffness matrix. For the second objective,

csq

C

is cost

per square meter,

t

C

cos

is initial cost of considered structural

elements, TOFA is summation of all floor areas. For the third

objective, vp is the average performance of visual perception,

FA

is the area of the façade of one floor, and SA is the

structural area that covers the same façade. From the standpoint

of constraints,

H

is the height of the high-rise model,

F

H

is

the floor height, and LS is the lease span distance between core

and the façade.

To calculate the first objective that is displacement, we

used the Karamba 3D parametric engineering plug-in [9],

which is developed for structural calculation and simulation in

Grasshopper 3D. Displacement, v value of the structure is

computed in the static way of calculation as follows [10]:

vFK ÷=

(9)

where

K

is the stiffness matrix, which is calculated by

Karamba 3D, is the lateral and gravity forces that are

implemented to the parametric high-rise model. In this respect

we considered wind loads as lateral load, live loads and dead

loads as gravity loads in order to assess the structural

performance of the model for each generated individual during

the evolutionary optimization process as follows [11]:

LDWF 6.12.1 ++= (10)

In order to calculate the wind load W, ASCE 07-5 standards is

also considered through the following equation:

))(613.0(

2

pdztz

GCIVKKKW =

(11)

where Wis the wind load in

2

/mKN ,

z

Kis the velocity

exposure coefficient,

zt

Kis the topographic factor,

d

K

is the

wind direction factor, Vis the basic wind speed, Gis the gust

factor, and

p

C

is the external pressure coefficient. We assume

that the building is located in a flat terrain with a basic wind

speed of

smV /48=

and exposure category B. To calculate the

dead loads, we assumed all structural elements and the weight

of the façade panels as follows:

)(

wtotwtotwtot

GCLBD ++= (12)

where

wtot

B is the total weight of beams generated in the high-

rise model,

wtot

CL is the summation of all columns used, and

wtot

G is the weight of glass elements used in the façade. To

calculate

wtot

B ,

wtot

CL and

wtot

G are calculated as follows:

()()

[]

{}

steelthhthwhw

dBBBBBBB ××−×−−×

ln

22 (13)

()()

[]

{}

steelthhthwhw

dCLCLCLCLCLCLCL ××−×−−×

ln

22

(14)

glasstharea

dGG ××

(15)

where

w

B is the width dimension of the beam,

h

B

is the height

of the beam,

th

B is the thickness of the beam,

ln

B is the total

length of beams used in the high-rise model.

w

CL

is the width

dimension of the column,

h

CL is height dimension,

th

CL is the

thickness of column section,

ln

CL is the total length of columns

used.

area

G is the total area covered by glass,

th

G is the

thickness of the glass that is assumed as 0.3 m.

steel

d and

glass

d

are the density values of steel and glass that we assumed as 77

and 25

3

/mkg . To calculate the live load, first we compute

2016 IEEE Congress on Evolutionary Computation (CEC) 2273

TOFA to multiply with live load per square meter as following

equation:

¦

=

=

FN

k

k

ATOFA

1

(16)

sq

LTOFAL ×=

(17)

where FN is the floor number of generated high-rise building,

k

A is the area of kth floor, and

sq

L

is the dead load per square

meter that is considered as 4

2

/mKN

As second objective, we calculate the cost per square

meter considering façade elements, structural elements, core,

substructure, interior divisions and mechanical, electrical,

plumbing services. This means that each initial cost that we

drive from the model is depending upon the generated high-rise

alternative. The formulation of initial cost is stated as follow

:

(

)

(

)

mepwsubscrslbstrfcdt

CCCCCCCC ++++++=

intcos

(18)

where

t

C

cos

is the initial construction cost of each generated

high-rise alternative,

fcd

C

is the cost of façade elements,

str

C

is the cost of structural elements such as beams and columns,

slb

C is the cost of slabs.

cr

C is the cost of core,

subs

C is the cost

of sub-structure,

w

C

int

is cost of interior walls used for space

divisions,

mep

C

is the cost of mechanical, electrical, and

plumbing services. First, we are starting to explain the

calculation the summation of

slbstrfcd

CCC ,,

for each high-rise

alternative as follows:

(

)

()

+×+×=

−−++ strustrfcdufcdslbstrfcd

CLCAC

()()

[]

slbu

CTOFATOFA

−

××− 25.0 (19)

where

fcd

A

is the façade area that covers each high-rise building

alternative,

fcdu

C

−

is the unit price of façade,

str

Lis the total

length of structural elements,

stru

C

−

is the unit price of

structural elements, FN is the total floor count of each

generated high-rise alternative,

k

A

is the area of kth floor,

TOFA is the total floor area, and

slbu

C

−

is the unit price of

slab. To calculate slab area without core spaces, we subtract the

total floor area and core area. In order to calculate the core area,

we considered to generate 25% of TOFA as average core area

based on implemented building projects for each high-rise

alternative [12]. Addition to this, we calculate the cost for

mepwsubscr

CCCC ,,,

int

considering average percentages for

mepwsubs

CCC ,,

int

as 8%, 9%, and 17% that are based on high-

rise economics [13] as follows:

+

×

=

−

+++

cru

mepwsubscr

C

TOFA

C25.0

int

»

»

¼

º

«

«

¬

ª

¸

¸

¹

·

¨

¨

©

§++×

−

−

−

−

−

−

11

int

1

17.009.008.0

mepu

wusubsu

C

CC

TOFA

(20)

where

mepuwusubsucru

CCCC

−−−−

,,,

int

are the unit prices of core,

substructure, interior walls, and mechanical /electrical/

plumbing services.

As third objective, we calculate the visual perception,

which is highly important for providing spacious, luminous,

and comfortable spaces. The main idea behind the formulation

of this objective is to maximize the glass area that provides the

visual perception from inside out. It is obvious that this

objective is conflicting with structural performance and initial

cost of the building. The visual perception requires less

structural elements for indoor spaces. This fact affects the cost

of structural elements due to less material usage, while

increasing the cost of façade design because of the increased

area of glass panels. This suggests finding a solution that is

capable to present desirable visual perception while having a

suitable performance for structure and acceptable price for the

initial cost of the construction.

In order to calculate this objective function, we subtract the

structural area from façade area of one floor. First, we compute

the total façade area by calculating the area of trapezoid

geometry for each side of the building that are named as side a,

b, c, and d. To do this, we considered two different height

values that are located in between top–middle and middle-

bottom parts, while considering each length dimension of each

rectangle located in bottom-middle-top parts to generate the

high-rise surface as follows:

+

»

»

¼

º

«

«

¬

ª

¸

¸

¹

·

¨

¨

©

§×

+

=

dcba

dcbadcba

Hmt

MLTL

FA

,,,

,,,,,,

2

»

»

¼

º

«

«

¬

ª

×

¸

¸

¹

·

¨

¨

©

§+

dcba

dcbadcba Hmb

BLML

,,,

,,,,,,

2

(21)

where

FA

is the façade area of the building,

dcba

TL

,,,

is the

length of a, b, c, and d sides of top rectangle geometry.

dcba

ML

,,,

is the length of a, b, c, and d sides of middle rectangle geometry.

dcba

BL

,,,

is the length of a, b, c, and d sides of bottom rectangle

geometry.

dcba

Hmt

,,,

is the height value between top and

middle rectangle parts for a, b, c, and d sides. And

dcba

Hmb

,,,

is

the height value between middle and bottom rectangle parts for

a, b, c, and d sides of the model. Secondly, we calculate the area

of structural elements considering the length of structure on the

façade to multiply with the height dimensions of columns and

beams as follows:

(

)

(

)

fhfh

CLCLBBSA

lnln

×+×=

(22)

where SA is the structural area on the façade of high-rise

model,

h

B is the height of the beams,

f

B

ln

is the length of

beams located on the façade,

h

CL is the height of the columns,

2274 2016 IEEE Congress on Evolutionary Computation (CEC)

f

CL

ln

is the length of columns on the façade. After the

calculation of subtracted value between area of façade and area

of structural elements, we divide this value with floor count to

gain average visual perception performance.

III.

GENERATIVE

HIGH-RISE

MODEL

In our case, we developed the layout of the high-rise

model based on a central core without any interior structure.

The main idea behind it is not only to support the building frame

for lateral loads, but also to keep the flexibility of the interior

spaces that affect profitability. The generative high-rise part of

the model consists of several steps, which can be expressed in

four parts as illustrated in Fig. 1.

Fig. 1. Generative steps of high-rise model

As the first step, we define the main controller points in

three different elevations namely bottom, middle, and top

zones. The reason for choosing three zones is to generate

flexible high-rise design alternatives. By doing so, the

computational optimization part has a chance to discover

distinct alternative feasible solutions. Secondly, we generate

rectangular shapes based on the main controller points. In

addition, rotation of middle and top rectangles are added as

decision variables in order to consider bended high-rise shapes.

Based upon the generated outer rectangles that address the main

boundaries of the building, we generate rectangles for obtaining

the building core with equations (19-20). In the third step, we

generate the 3D geometry of the high-rise design by using a

swept surface in order to define a floor number through division

in the fourth step. Finally, we generate diagrid structure with its

specific parameters in order to control the repetitions on

horizontal (xy) and vertical (z) planes.

IV.

H

EURISTIC

O

PTIMIZATION

A

LGORITHM

S

A.

Multi-Objective Non-Dominated Sorting Genetic

Algorithm

In this paper, we deal with designing high-rise building

problem by using improved version of NSGA (Non-Dominated

Sorting Genetic Algorithm), which is called NSGA-II (Non-

Dominated Sorting Genetic Algorithm II). The NSGA II is a

Multi-Objective Evolutionary Algorithm (MOEA), which is

developed by [14]. NSGA-II is an elitist, multi-objective GA

that can also handle constraints. The algorithm builds on and

improves the earlier NSGA algorithm. It is acknowledged that

NSGA-II is able to address difficult real-world multi-objective

problems, and for which it may achieve a homogenously

distributed set of Pareto-optimal solutions. The key features of

the algorithm are as follows:

• Non-dominated sorting of solutions, using a fast O (MN2)

sorting algorithm.

• Parameterless diversity calculation mechanism based on the

cuboid volume between neighboring elements of the same rank,

a measure termed “Crowding Distance”.

• Diversity-preserving binary tournament selection, based on

comparison of constraint violation, ranking and crowding

distance.

• Diversity preserving elitist approach that combines elite

parent and offspring members taking into account constraint

violation, rank, and uniqueness.

• Simulated Binary Crossover genetic operator [14].

• Polynomial mutation operator [14].

The algorithm starts by initializing a random population of

individuals. Since we have as decision variables, the solution

representation for an individual is in fact the decision variables

in 16 dimensions of the problem formulated above. In other

words, for the first dimension of the first individual and so on.

The sequence of steps that the NSGA-II algorithm performs at

each generation is as follows:

1. The parent population is sorted according to each of its

member’s non-domination. Through this process, each solution

(population member) is assigned a value that represents its rank

within the population. NSGA-II achieves a fast sorting of

population members, through extensive bookkeeping, namely

through associating the number of solutions dominating, and

references to solutions that are dominated by any population

member. This procedure places the population members to

discrete sets according to their Pareto ranking.

2. For each of the population members, the crowding distance

is calculated. The crowding distance is calculated for members

of each rank separately, and, for each of the members,

comprises of the distances to their nearest neighbors, summed

overall objective function dimensions.

3. In this step, the mating pool is formed; individuals are

selected from the parent population following a binary

tournament, until the mating pool is filled. During the

tournament selection, the constraint violation of the solutions is

firstly compared. In case both solutions are violating, the one

with the least violation is selected. In case one is violating, the

other is selected. In case none are violating, non-dominating

ranks are compared. The solution with the lower rank among

the two is picked. In case they are non-dominating, crowding

distance is finally compared, and the one with the highest

crowding distance is selected.

4. The mating pool individuals are subjected to the genetic

operators, crossover through SBX and polynomial mutation, in

order to form the next generation.

5. The elitism step is performed. In this step, the current and

previous populations’ members are merged in different pools

according to their non-dominance, and each of those is sorted

with members having the highest crowding distance coming

2016 IEEE Congress on Evolutionary Computation (CEC) 2275

first. The elitist population is formed by including as many of

the first rank individuals as possible for the population size; if

there are spaces left, the second rank follows and so on [15].

The details of the NSGA-II algorithm can be found in [14].

B.

Multi-objective self-adaptive differential evolution

algorithm

This section first briefly review several DE algorithms

developed for real-parameter constrained optimization

problems in recent years. DE is firstly proposed by Storn and

Price [16, 17]. A recent survey can be found in [18]. DE is a

very powerful algorithm for real-parameter optimization

problems. There have been several DE algorithms combined

with different constraint-handling techniques, which can be

found in [19-33]. For more related works and discussions, we

refer to the survey in [29]. From the literature review, we can

see that DE algorithm is an efficient algorithm for constrained

optimization problems. Regarding the multi-objective DE

algorithms, we refer to [34, 35]. For this reason, we choose a

DE variant called jDE algorithm in [36] to solve the problem

on hand in this paper.

In the traditional DE algorithm, the initial target population

has NP individuals having a

D

-dimensional real-valued

parameter vectors. Each vector, also known as chromosome,

keeps an alternative solution to the multidimensional

optimization problem. Each vector is obtained randomly and

uniformly within the search space restricted by the prescribed

minimum and maximum bounds

()

maxmin

,

ijij

xx . Thus, the

initialization of

th

j

component of th

i

vector can be defined as:

()

rxxxx

ijijijij

×−+=

minmaxmin0

(23)

where

0

ij

x is the th

i

target individual at generation

0=g

; and

r

is a uniform random number in the range

[]

1,0

.

Mutation is the first step to get new solutions. Nevertheless,

it consists of changing the value of parameters based on the

difference of two vectors. The difference vector based mutation

operator is believed to be one of the main strength of DEs [15,

16]. These differences tend to adapt to the natural scaling of the

problem over generations. Hence, DE differs from other

evolutionary algorithms since it relies on a difference vector

with a single scale factor

F

for all variables.

In order to obtain mutant individuals, the weighted

difference of two individuals from target population is added to

a third individual randomly selected from population as

follows:

()

1

1

1−

−

−

−×+=

g

cj

g

bj

g

aj

g

ij

xxFxv (24)

where cba ,, are three randomly chosen individuals from

the target population such that ),..,1( NPicba ∈≠≠≠ and

Dj ,..,1=

. 0>F is a mutation scale factor influencing the

differential variation between two individuals as follows:

otherwise

DjorCRrif

x

v

u

j

g

ij

g

ij

g

ij

g

ij

=≤

°

¯

°

®

=

−1

(25)

where the index

j

D

is a randomly chosen dimension from

),..,1( Dj =

. It makes sure that at least one parameter of the trial

individual

g

ij

u will be different from the target individual

1−g

ij

x.

CR is called the crossover rate and appears as a control

parameter of DE just like

F

. CR is a user-defined crossover

constant in the range

[]

1,0

, and

g

ij

r

is a uniform random number

in

[]

1,0

.

When trial individuals are generated, parameter values

might violate search ranges. For this reason, parameter values

violating the search range are randomly and uniformly re-

generated as follows:

()

rxxxx

ijijij

g

ij

×−+=

minmaxmin

(26)

The next step of the algorithm is selection. For the next

generation, selection is normally based on the survival of the

fittest among the trial and target individuals such that:

otherwise

xfufif

x

u

x

g

i

g

i

g

i

g

i

g

i

)()(

1

1

−

−

≤

¯

®

= (27)

Selection of the control parameters of

D

E

is a serious

task. In the DE algorithm above, a novel self-adapting

parameter scheme developed by [36] was employed, so called

jDE. It is very simple, effective and converges much faster than

the traditional DE, particularly when the dimensionality of the

problem is high or the problem concerned is complicated. In

jDE, each individual has its own

i

F and

i

CR values. Initially,

they are assigned to 5.0=

i

CR and 9.0=

i

F, and they are

updated as follows:

otherwise

trif

F

FrF

F

g

i

ul

g

i

12

1

1

<

¯

®

×+

=

−

(28)

otherwise

trif

CR

r

CR

g

i

g

i

24

1

3

<

¯

®

=

−

(29)

where

{}

4,3,2,1∈

j

r

are uniform random numbers in the range

[]

1,0

.

1

t and

2

t denote the probabilities to adjust the

F

and

CR . They are taken as 1.0

21

== tt , 1.0=

l

F and 9.0=

u

F.

The above jDE is designed for single objective

unconstrained/constrained real-parameter optimization

problems. In order to extend it to multi-objective constrained

optimization problem as in this paper, we mainly use the non-

dominated sort algorithm of Deb [14]. To achieve it, first, we

combine the target and trial population to apply a fast-non-

dominated sort algorithm of Deb [14], in which a rank is

assigned to each individual in the combined population. Then,

2276 2016 IEEE Congress on Evolutionary Computation (CEC)

we calculate the crowding distance for each individual in the

combined population. Finally, we determine the population for

the next generation through the use of Crowded-Comparison

Operator considering constrained-domination principle in [14].

For details, we refer to Deb [14]. The flowchart of the multi-

objective jDE algorithm is given in Fig. 2.

Fig. 2. Multi-Objective jDE procedure

V.

COMPUTATIONAL

RESULTS

AND

DISCUSSION

The NSGA-II and jDE algorithms are run on a computer

with Intel I7 4 core processor at 2.7 GHz, 16 GB DDR3

memory, and 256 GB solid-state drive. For both algorithms, we

took the population size as 200. The algorithm has terminated

after 250 generations. The average computation time for each

generation was 24.3 minutes. Thus for 200 population size and

250 generations as a termination criterion took 81 hours, or

3.375 days.

In order to analyze the performance of algorithms, we

employ the hyper volume (HV), measuring the volume of the

dominated portion of the objective space. For every 25-

generation, we recorded the Pareto front of each algorithm and

compared to each other by using the HV indicator (the higher

is better). As seen in Fig. 2, jDE yields much better HV values

than NSGA-II for most generations. It can be seen from Fig. 7

that at 200 generations, both algorithms reached at their best

values. However, the performance of jDE is better than NSGA-

II because of the higher HV value yielded in terms of the

standpoint of engineering discipline.

For the high-rise design problem, we considered two

Pareto-fronts obtained from NSGA-II and jDE in order to

analyze them in terms of architectural aspects. In order to make

the visual inspection to determine the architectural features

discovered by different algorithms, we considered 200th

generation that presents the highest HVs for both NSGA-II and

jDE presented in Fig. 3 and Fig. 4.

Fig. 3. Hypervolume plot for NSGA-II and jDE

From the point of NSGA-II, cost values in the Pareto front

observed in between 1498.73 TL and 1728.78 TL. On the other

hand, jDE discovered cost values between 1490.22 and

1597.04. With regard to performance of displacement, NSGA-

II presented results between 0.10m and 0.33m, while jDE

suggested 0.14m and 0.38m. As final objective that is the

performance of visual perception, NSGA-II presented results

between 635.55m and 1544.01, while jDE discovered

1136.08m and 1729.48m. According to this information, it is

possible to state that the advantage of NSGA-II is based on the

2016 IEEE Congress on Evolutionary Computation (CEC) 2277

cost objective due to wide range of alternatives. However, jDE

generated solution alternatives that have wider range for

displacement and visual perception objectives than NSGA-II.

This is why HV of jDE algorithm is higher than the NSGA-II

algorithm.

Fig. 4. NSGA-II results of 200th generation

Fig. 5. jDE results of 200th generation

Fig. 6. Selected solutions of NSGA-II from 200th generation

Fig. 7. Selected solutions of jDE from 200th generation

From the standpoint of the displacement and visual

perception objectives, we can clearly observe that the decision

variables (dimensions) of structural elements have an impact on

the performance of displacement and visual perception.

However, based on our observation, it seems that the decision

variable, which is the structural division on z-plane, is much

more important for displacement and visual perception

objectives. We can clearly see that division count on z-plane

discovered by NSGA-II is bigger than the one found by jDE.

This observation causes to bigger displacement value and

higher visual perception discovered by jDE. This fact can

explain why the displacement of jDE is bigger than NSGA-II,

while the visual performance examined by jDE is also better

than NSGA-II. In addition, we also think that discovered

solution alternatives having more division on z-plane affects the

cost objective negatively.

Furthermore, NSGA-II solution alternatives have bigger

rotation-angle for the top-zone than those found by jDE,

whereas rotation angle of the mid-zones discovered by both

algorithms are similar. This is another observation that

decreases the wind load on the building façade. Even though

the results of NSGA-II present better displacement performance

and wider range for cost objective, jDE results can be seen as

more desirable solution alternatives in terms of cost and visual

perception objectives due to smaller cost and bigger spaces. It

is obvious that different optimization algorithms present

different design alternatives for decision makers with wide

ranges of each objective function. The novelty of this paper

stems from the fact that it considers several heuristic

optimization algorithms in the conceptual phase of design

process, which are not very well studied in the literature. By

doing so, we are able to handle the same design problem

through different heuristic optimization methods. This gives an

advantage to observe different relationships between solution

alternatives for the Architect and/or decision maker.

2278 2016 IEEE Congress on Evolutionary Computation (CEC)

VI.

CONCLUSION

A high-rise form-finding building problem to reach

suitable solutions for hard and soft objectives is presented in

this paper. The problem is formulated as a multi-objective real-

parameter constrained optimization problem for the conflicting

objectives subject to several design constraints. To tackle the

problem on hand, we developed two different optimization

algorithms, so called NSGA-II and jDE algorithms in order to

obtain Pareto fronts with diversified non-dominated solutions.

The extensive computational results show that the jDE

algorithm yields much more desirable Pareto front than the

NSGA-II algorithm. To the best of our knowledge, this is the

first reported application of multi-objective evolutionary

algorithms for high-rise form finding problem focusing on the

conceptual phase in the literature.

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